Outside the vet's surgery on mock-Tudor Chester Road in the eastern suburbs of Birmingham, a sign offers pets a "£99 booster for life" jab. Does Britain's self-effacing second city need a similar boost in the shape of a Ken-or-Boris executive mayor? That is the question the city's 700,000 voters will be asked in a referendum on local election day, 3 May.

The rejuvenation of Birmingham's city centre, botched in the 1960s, is an acknowledged commercial and cultural success. After a nasty recession, Jaguar's Castle Vale factory in Chester Road is booming in the world's luxury markets. So is Land Rover, now also owned by Tata of India, just across the city line in Solihull. The headache for unions now is compulsory weekend shifts, rather than layoffs. The Jaguar XF Sportbrake will be built at Chester Road, it was confirmed when the car was unveiled at the Geneva motor show this month. Land Rover is building a new plant in Wolverhampton.

Hi-tech, science-driven and digital: there is much to celebrate. The New York Times recently placed Brum at 19 in its list of "45 places to go in 2012". But not even that flattery assuages anguish over the local skills shortage (40% of pupils leave school without five decent GCSEs), stubbornly high youth unemployment, or over the notorious transport gridlock and the feeling that Manchester has managed to overhaul bigger Birmingham just like its football teams. It even handled its 2011 riots better.

Prof Wyn Grant of Warwick University, who has been chairing a study into the case for and against elected mayors, to be published next month, says: "Birmingham has not marketed itself very well as the second city. It's taken a lot of knocks since the 60s and the revitalisation of the city centre has not lifted morale and confidence."

Desperate

Dr David Hardman, who runs the city's university-linked science park, puts it succinctly. "What a mayor would have to do by 2026 is create something in Birmingham which makes people get off the train there instead of going on to Manchester."

May's ballot on devolving power from Whitehall has been imposed by Whitehall on Birmingham and nine other English cities, from Newcastle and Leeds to Manchester and Bristol. Like Labour before it, the coalition government is desperate to find a more dynamic and responsive form of municipal leadership to revitalise – and rebalance – regional economies. Despite lurid efforts by the No camp to make an elected mayor sound like a tyrant – a Brummie Nero or Genghis Khan – the campaign has so far barely fizzled into life beyond the chamber of commerce and the universities, "just business and the intelligentsia," Yes boosters admit.

Chester Road is part of the Erdington constituency held until 2010 by the leading mayoral contender, Labour's Siôn Simon, who took a gamble and gave up his seat to campaign for mayor. Yet, just eight weeks before the referendum, few local voters know anything about it. "I've seen small bits in the newspaper and on TV. I don't usually vote, but I might for a mayor. If London's got one and it's good for business, I don't see why the second city shouldn't have one too," says Kevin Reddan, 31, a self-employed engineer.

Reddan is the exception. Young mums and grandads with prams, middle-aged men walking their alsatians, none of them have heard a word. "Don't know anything mate, and I don't care," says a voter cleaning his car. Bustling New Street in the city centre's pedestrianised shopping quarter is no better informed.

In multi-ethnic Aston ward, close to Spaghetti Junction and one of the city's poorest, the message is the same. "No, we haven't heard," admit a neat Kashmiri couple in Grosvenor Road, though the man guiding a delivery lorry into nearby Al-Amin Cash & Carry crisply replies "American-style mayors" and gives a thumbs-up. Aston's Lib Dem councillor and wannabe regional police commissioner, Ayoub Khan, is personally undecided: Birmingham's black and ethnic minorities (20% south Asian, 7% black) are both confused and ill-informed about a mayor, he admits.

Yet the Yes camp is doggedly convinced it will prevail here on 3 May despite apathy and ignorance which many blame on Birmingham city council's hostility to a Whitehall-dictated blow to the status quo and weak regional media. In 2001 a consultative referendum voted by 56% to 45% for a mayoral system, with voters divided between two versions. Councillors decided to ignore both the vote and the grumbling about the failure to attract inward investment to replace lost manufacturing in metal-bashing Castle Vale and elsewhere.

Such complaints persist: hence the Yes camp's cautious optimism and the No camp's less convincing "the result may be close". In truth, while minor parties from Ukip to the Greens are solidly opposed, all three main parties are split on the issue. Solid but uncharismatic Mike Whitby, who has led a Tory-Lib Dem coalition at the Council House on Victoria Square since 2004, has modified his hostility and is widely expected to stand at the November election if Brum says yes.

His Labour predecessor, the veteran Sir Albert Bore – no media-savvy Boris Johnson or Ken Livingstone either – has already thrown his hat into the ring, though he expects to be council leader again if Labour gets its expected win on 3 May. So has Gisela Stuart, Labour MP for suburban Edgbaston, which she holds against the odds. The former CBI boss and (briefly) Labour industry minister, Lord (Digby) Jones, the only big Brummie personality with a national profile in sight, has ruled himself out. John Hemming, a leading No campaigner and Yardley's maverick Lib Dem MP, has not quite done so, despite exotic political and personal baggage that can match a Ken or Boris.

Liam Byrne, a Labour former cabinet minister whose Hodge Hill seat is set to go under boundary changes, is hovering on the fringe, waiting to see if voters back the mayoral principle. What with squabbling over tactics (does Simon have privileged access to voter records? asks Stuart) and manoeuvring (might Bore and Ed Miliband swing behind Byrne?), it has been uninspiring.

Labour will ballot its 3,000 to 4,000 members in the city on the party's choice of candidate in June. Even allowing for the uncertainties of a second preference ballot, Labour's choice may be the crucial one, since its candidate would be expected to win. All have flaws, but Simon is the tipped favourite.

Simon, who thinks the city is run by a dysfunctional, introverted clique of councillors and officials, launched a 10-point revitalisation plan in an Afro-Caribbean food warehouse this week. Bore has a rival plan. But Miliband's equivocation on mayors (Labour opposes Whitehall-imposed referendums) may be an issue because his own constituency base of Doncaster will ballot on 3 May to ditch its brief but turbulent experiment with an elected English Democrat mayor. It does not help.

Enhanced

Privately, Whitby asks the crucial question: "If you just change my title and directly elect me, that's pointless. If you give me new powers and directly elect me, that might be worth it." If a high-profile city mayor could bang heads together – a bridge-builder across public and private, corporate and academic, Whitehall and venture capitalist sectors – with enhanced powers, it could make a difference, London-style. It is precisely the complaint that Brum's mayor will not have such powers over the capital's 32 lesser councils that prompted Lord Jones to say No. Jones wants a mayor, not just for the city, but with strategic regional powers over the neighbouring and rival Black Country boroughs such as Wolverhampton and beyond.

In negotiations with local government chiefs, Eric Pickles and his deputy, Greg Clark, have hinted at extra powers and extra cash: they bribed Liverpool council into embracing the mayoral model without a referendum. But they remain vague on details.

Councillors with long memories recall that Margaret Thatcher abolished the West Midlands metropolitan county council and its kind in 1986 and that Pickles announced the abolition of the Advantage West Midlands regional development agency and other RDAs in 2010. Their replacement, local enterprise partnerships (LEPs) are deemed too weak. Council chiefs are therefore wary, just as ministers fear granting the London mayor's powers to other ambitious city-regions. Wolverhampton and Solihull look at Birmingham and are nervous too.

By general consent voters do not care much about local government structures, but can distinguish success from failure. As someone said of Welsh and Scottish devolution, it is "a process, not an event". It is a thesis Alex Salmond is testing to the limit in Edinburgh. Let's have a mayor and watch him prove the post can work, then gently extend his soft power beyond the city boundary, the optimists argue. It's a gimmick that puts too much power into one "power freak's" hands, reply the wary. Staid status quo or shiny experiment? Across Britain, it is a leap in the dark.